| L(s) = 1 | + 4.86·2-s − 3·3-s + 15.6·4-s − 5·5-s − 14.5·6-s + 19.6·7-s + 37.3·8-s + 9·9-s − 24.3·10-s − 47.0·12-s − 1.18·13-s + 95.6·14-s + 15·15-s + 56.2·16-s + 9.08·17-s + 43.7·18-s + 39.4·19-s − 78.3·20-s − 58.9·21-s + 13.8·23-s − 111.·24-s + 25·25-s − 5.77·26-s − 27·27-s + 307.·28-s + 226.·29-s + 72.9·30-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.95·4-s − 0.447·5-s − 0.993·6-s + 1.06·7-s + 1.64·8-s + 0.333·9-s − 0.769·10-s − 1.13·12-s − 0.0253·13-s + 1.82·14-s + 0.258·15-s + 0.878·16-s + 0.129·17-s + 0.573·18-s + 0.476·19-s − 0.876·20-s − 0.612·21-s + 0.125·23-s − 0.952·24-s + 0.200·25-s − 0.0435·26-s − 0.192·27-s + 2.07·28-s + 1.45·29-s + 0.444·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.044560178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.044560178\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.86T + 8T^{2} \) |
| 7 | \( 1 - 19.6T + 343T^{2} \) |
| 13 | \( 1 + 1.18T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.08T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 33.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 835.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 53.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 634.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 606.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 397.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 177.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 588.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.661108210156907218850588774501, −7.87330995884703647885322716217, −6.96500329463267031363541180667, −6.36119785898695580502024247364, −5.28729119754542927647441942410, −4.92382751308099649147262952258, −4.15358907673099241000210744234, −3.24466591004075340077497158044, −2.16648994633575922893842663821, −0.952166874672717127990660455847,
0.952166874672717127990660455847, 2.16648994633575922893842663821, 3.24466591004075340077497158044, 4.15358907673099241000210744234, 4.92382751308099649147262952258, 5.28729119754542927647441942410, 6.36119785898695580502024247364, 6.96500329463267031363541180667, 7.87330995884703647885322716217, 8.661108210156907218850588774501
